The Incommensuration Theorem

Statement of the Intrinsics of Comparison

1R1 Where discussing the concept of comparison in the common usage terms (i.e., where holding the terms subjective and objective as implicit), it is apparent that exactly four other concepts are both necessary and sufficient for the formal consideration of the concept of comparison. These concepts are 'sameness', 'difference', 'content', and 'context'. Any process of comparison involves all four of these terms as intrinsics. Where there is comparison there must be sameness, difference, content, and context. No comparison can be defined without implicitly making reference to all six of these concepts. As such, the collection of these six abstract concepts will be hereafter referred to as 'the intrinsics of comparison'.


Concept Inseparability

1R2 In consideration of the meanings of the intrinsics of comparison, certain statements of inseparability are apparent. For example, it is clear that the concepts of sameness and difference are inseparable. Whenever one appears, the other is implicit. Where there is sameness, there must also be difference. Where there is difference, there must also be sameness.

1R3 Further, the concepts of content and context are inseparable. Where there is content, there must also be context. Where there is context there must also be content.

1R4 This inseparability of the concepts of content and context, and the concepts of sameness and difference, is assumed as being fundamental to, and irrevocably inherent in, all discussions of the concept of comparison, or of anything that is defined in terms of comparison.

1R5 Where the above concepts of comparison, sameness, difference, content, and context, are inseparable, it is also clear that they are distinct concepts with distinct meanings. For example, the meanings of these terms are not interchangeable in any formal statement, without changing or altering the fundamental meaning of that statement.

1R6 It is understood that similar aspects of distinctness and non-interchangeability will apply similarly to any terms that are defined in terms of these.


The Mutual Applicability of the Intrinsics

1R7 It is clear that some of the intrinsic concepts of comparison are applicable to one another. In particular, the concepts of sameness and difference can be applied to the concepts of content and context. Also, (the concepts of) content and context can be applied to (the concepts of) sameness or difference. As it turns out, there are four primary concepts which are formed from these specific applications (i.e., as second order compounds and conjugations).

1R8 Where there is a sameness of subjective context, the following definitions hold about the objective.

continuity is a reference to a sameness of content where there is a sameness of context,

discontinuity is a reference to a difference of content where there is a sameness of context,

symmetry is a reference to a sameness of content where there is a difference of context,

asymmetry is a reference to a difference of content where there is a difference of context,


On the Formation of the Tertiary Compounds

1R9 Clearly the concepts of symmetry and continuity are not the same, for 1) there are discrete and non-continuous things which are symmetric, and 2) twisted continuous things that are non symmetric. For this reason, it is valid to consider the formation and inter-applicability of these compound concepts of the intrinsics of comparison, and their opposites, to one another. However, in considering the mutual compatibility and applicability of the secondary compounds to form tertiary conceptual compounds, certain conflicts become immediately apparent.

1R10 For example, some of these tertiary compound candidates can be rejected since the duplicate application of the same concept to itself is of no semantic value. In this way four potential candidates are discarded from the list of potentially valid mutual applications.

1R11 Another of these conflicts concerns the mutual applicability of the types of comparison to one another in the terms of 'opposite meanings'. Some of the pairs of these concepts are mutually incompatible as they are clearly opposites of one another. For example, the concept of continuity with the concept of discontinuity asserts that for the same context, something is both identically the same as and identically different than itself. Clearly something (in this case content) cannot both be the same and be different in the same way at the same time.

1R12 In the same way that the formation of a single compound concept out of 'sameness of content' and 'difference of content' would be invalid as it violates concept distinctness (as defined earlier). A compound that assumes that the context is both the same and different violates distinctness. For example, it would be problematic to assert that for the same content, its context is both the same and different. Thus, there is the limitation (in the formation of compound concepts of the four types of comparison) that something cannot be both the same and different in the same way at the same time.

1R13 Insofar as there are two pairs (of the four concepts which are available) which are opposites to one another, then clearly they will not be mutually compatible or applicable. In this way two additional potential third compound candidates can be dropped from the acceptable list of mutual applications. Other potential candidates will also be dropped where they violate the requirement of distinctiveness.

1R14 In addition, due to the inseparability and distinctiveness of the intrinsic concepts of comparison themselves, the formation of the compounds of using these are subject to certain limits. In particular, one of these limits is 1) where the concept of content appears, the concept of context must appear also, and 2) where the concept of context appears, the concept of content must also appear. This, however, is not a problem as the definitions of the terms always ensures that there will be an even matching of a content with a context and a context with a content, wherever these appear.

1R15 However, in a very similar manner (as in the immediately above paragraph), two more candidates for tertiary compound concepts must be rejected. This is a more subtle limitation. In the same manner that the concepts of content and context are inseparable, conceptually distinct, and non-interchangeable, the concepts of sameness and difference are also inseparable, distinct, and non-interchangeable. In particular, where the concept of sameness appears, the concept of difference must appear also, and vise versa. The 'number of appearances' of each concept must match the number of appearances of its opposite.

1R16 Fortunately, all of these limitations can be accounted for fairly easily by noting that when any one of the concepts in each of the concept axis domains appears, its mate must appear also. All that is needed to determine whether the third order compounds of the intrinsics of comparisons are valid is to check that all of the concepts 'add up' or 'cancel out' under the inseparability relationships. In this way, the four compounds formed by the four types of comparison can be checked rather readily.

1R17 In summary form, the above limitations could be organized into three basic sets: where [!x] indicates an invalid application to self, [!y] indicates an invalid mutual application of opposites, and where [!z] indicates an invalid assumption of the separability of sameness and difference. As such, the results of the consideration of all candidates for the formation of the third compounds of the intrinsics of comparison may be listed as follows.

Table of Tertiary Conjunctions:

[!x] continuity and continuity (ss & ss)
[!z] continuity and symmetry (ss & sd)
[!y] continuity and discontinuity (ss & ds)
continuity and asymmetry (ss & dd)
[!x] symmetry and symmetry (sd & sd)
symmetry and discontinuity (sd & ds)
[!y] symmetry and asymmetry (sd & dd)
[!x] discontinuity and discontinuity (ds & ds)
[!z] discontinuity and asymmetry (ds & dd)
[!x] asymmetry and asymmetry (dd & dd)

1R18 From the above list, it is clear that there are only two remaining combinations which are mutually compatible; 1) continuity with asymmetry and 2) symmetry with discontinuity. All other combinations are, for various reasons, incompatible with one another.


The Incommensuration Theorem

1R19 Concept inseparability and the associations as defined, require that an eventity/comparison is either 'continuously asymmetric' OR it is 'discontinuously symmetric'. An eventity/comparison cannot be 'discontinuously asymmetric' nor 'continuously symmetric'. This constitutes the fundamental result of this essay, and is called the 'Incommensuration Theorem':

The concepts of symmetry and continuity cannot both be simultaneously and fundamentally applied to any eventity/comparison.

The concepts of asymmetry and discontinuity cannot both be simultaneously and fundamentally applied to any eventity/comparison.

1R20 Any absolute application of the concept of comparison must ultimately be continuous and asymmetric, or symmetric and discontinuous.

The Nature of Symmetry

2M The concept of symmetry is about invariance of content with transformations of context. To make this abstract definition a little easier to understand, consider the following metaphors and examples.

2M2 Consider a black square drawn on an otherwise blank page. Imagine that the paper is now rotated so that the top becomes the bottom. In this transformation, flipping the square end over end, the context of the square has changed. Imagine the perspective of the square looking out at the room. In this view, the ceiling becomes the floor and the floor becomes the ceiling. The context of the square has changed, but the square remains visually unchanged. The content remains the same, but the context has changed.

2M3 Consider the basis of any scientific theory (a knowledge of this world). The fundamental laws are all derived from, and based upon, the concept of symmetry. For example, the scientific experiment of measuring the boiling point of water will discover that it is 100 degrees Celsius. The results of this experiment do not depend on whether it is done somewhere in the USA or somewhere in France. It is reasonable to expect to get the same result regardless of the location in which the experiment was performed. The content (the result of the experiment) is the same even though the experiment is done in different places (a different context). The universality of the 'natural laws of science' is thus an implication of the concept of symmetry.

2M4 2M5 As another example, consider something which is in a closed and sealed bottle for one day. With the passage of time, the bottle is still the bottle, and the materials inside will remain there. There persists the same amount of material, even though the context of time has changed. This symmetry of material content in the context of time is the law of conservation of matter. The basic physical laws of conservation of matter and energy are symmetry laws. The very dynamics of the equations that serve as the foundations of technology, are all ultimately based on concepts of symmetry.

The Nature of Continuity

2M6 To assert continuity is to state that an infinitesimal change in context will always and necessarily result in an infinitesimal change in content. Continuity is where the content of a change is not (is never) 'drastic' or sudden, given any arbitrarily slight change in the context. It is an assertion that changes in content are only partially sensitive to changes in context.

2M7 For example, in ordinary mathematics, a "continuous function" is one where the curve is everywhere connected to itself. A discontinuous curve refers to a function that will make a jump from one value to another, with no intermediate steps. Such a curve is disconnected. One could describe a discontinuous function by saying that there exists an infinitely small region along the X axis (a sameness of context), where the value of the function (the content) changes abruptly. The ratio of change of content over change of context is infinite, indicating a break. Discontinuity is an abrupt shift of content while the context is the same.

2M8 To consider a continuity of self is to assert a wholeness to the self; that there is no part of self that is disconnected from, or rejected by, another part of self. To say that one has continuity of self is to essentially be a healthy, whole integral being.

2M9 Consider the work of a therapist. The main job of the therapist is to help the patient to know and accept all parts of self. A good therapist helps one to nurture and love oneself fully (in all aspects). This may require learning how to live well, and how to coordinate and balance all personal behaviors, beliefs, feelings, and attitudes. The primary job of the therapist is to get one to love, nurture, and accept all aspects of oneself into an integrated and healthy functioning whole.

2M10 Consider a phone conversation where one person abruptly, with no indication or forewarning, hangs up. One second, one is pleasantly involved in a conversation and in the next, silence. There is no significant environmental difference between the instant when the conversation was alive and the instant that it was not (i.e., the context is still much the same). The infinitesimal period of time that it took for the disconnection to happen (a sameness of context), is not in itself different, even though the content (the conversation) has shifted a lot (from being to not being).

2M11 To recognize how implicit this idea of continuity is in a culture, consider the use and language of the many ideas deeply held within religion, mysticism, and spirituality. Some of the deepest values, mythos, and beliefs relate to how one connects to deity, spirit, and the land. Continuity in a culture is about how one connects to one's ancestors and heritage through shared practices and rituals (rites of passage).

2M12 Quality and feeling are best considered in terms of continuity. For example, one could think of a spectrum of colors. Each color fades into the next. There is no boundary between them, no discrete place where one begins and another ends. All colors have a common spectrum and come from a common energy. The commonality is a form of continuity. Additionally, they are all part of the electromagnetic spectrum, which is always a continuous field of energy without exact boundary.

Law and Locality

Although most of the examples for continuity have an inherently subjective nature, this is not to imply that the concept of continuity and of discontinuity are absent from formal consideration within the objective physical sciences. The quantum theory, for example, considers an inherently non-continuous process called 'the quantum jump'.

There is a parallel between the incommensuration theorem and the Bell Theorem of physics. In essence, the Bell Theorem states that any physical theory of reality cannot both assert that reality is "lawful" and that reality is "local". According to the Bell Theorem, reality can be either completely lawful and somewhat non-local or it can be completely local, and somewhat non-lawful. The concepts of lawfulness and of locality can be regarded as special cases of the more general concepts of symmetry and continuity (respectively).

It is an inherent assumption of science that, given similar environmental conditions, the results of a physical procedure performed in one place and time will be the same as the results of the same procedure implemented in other places and times. This constancy of the results of empirical experiments (a sameness of content) performed in different times and places (a difference of context) is an expression of the concept of symmetry. Science assumes that the essential nature of the dynamics of physical process is everywhere the same in the universe.

Science regards the 'laws of physical process' as invariant under transformations of changing times and position. In that the methodology of science itself depends on the notion of repeatable observations, the scientific method itself inherently involves an implicit assumption of symmetry. As such, the mathematical expression of the laws of nature (as discovered by the methodology of science) are all based on notions of symmetry. For example, the conservation of matter and energy is a symmetry law. The foundations of the theories of relativity are based on ideas of a sameness of various types of relationship under circumstances of changing position and momentum. The ultimate assertion of symmetry of science is to assert that the same physical laws (as content) apply everywhere (an invariance) in the universe (a context). As such, for science to regard the universe as lawful is to assert a fundamental notion of symmetry (1).

The notion of locality as used within science is essentially an assertion that no physical influence, interaction, or signal can travel faster than the speed of light. To assert locality is to assert that all physical process involve dynamics which do not have instantaneous transits across space (a jump between arbitrarily separated points in zero time). If an interaction spans any distance, then it must also span some nonzero duration.

The metrics of time and space are contextual metrics, and interaction and substance (physical matter) is regarded as being a 'content' within that context (2). This notion that there is no influence, interaction, or signal (all of which are content) which can instantaneously cross an arbitrary distance of space (a context) is equivalent to the notion of continuity. For any small change (a process, interaction, or signal, as a content) there must be a corresponding change in the context of that interaction (in time and space). The notion of locality, in asserting that there can be no abrupt instantaneous changes, is a special case of the concept of continuity (3).

However, the incommensuration theorem regards symmetry and continuity to be mutually inconsistent. They cannot be simultaneously applied to the same immanent modal identity. A theory of reality is a theory of being, and thus must be formulated from an immanent, rather than omniscient, basis. Therefore, the notion that reality is absolutely lawful is inconsistent with the notion that reality is absolutely local. In effect, a conception of reality must either allow 1) multiple mutually inconsistent sets of different physical laws which are applicable in different places and times (failure of lawfulness and symmetry), or 2) inherently non-local interactions and changes which are fundamentally non-deterministic (failure of locality and continuity). No theory of reality can assert both total generality, and total determinism.



Notes:
[1] Any aspects or process of the physical world which are real and yet which are inherently not repeatable (or observable) would be non-lawful and non-symmetric in time and space and therefore ultimately outside of the scope of study available to the method of science. To assert that something is real is different than asserting that it exists or that it is objective.

[2] Technically, it is incorrect to identify the concept of interaction directly with content. By the root tautology, the notion of interaction is equivalent to comparison, and therefore both content and context are to be regarded as intrinsic aspects of interaction. Thus, neither content nor context can be identified as interaction and comparison itself. The concept of change (in the sense of physical process), however, can be directly regarded as a content, thus restoring the logic of this essay.

[3] To assert absolute locality is to assume a total continuity of interaction at all scales, down to and including the (microscopic) scale of absolute zero in both distance and duration. It is to require that theories of physical reality are defined in terms of deterministic law rather than in terms of causal law. However, the scientific method can only make observations about causality. It inherently cannot (even in principle) make any direct observation or assertion about continuity or locality. The scientific method cannot make any direct observation about symmetry either, even though it must implicitly assume an inherent symmetry in reality in its practice. No single experiment could ever possibly validate the absolute and universal truth of either continuity (the all interaction and change is local) or symmetry (that reality is lawful).



Consistency and Completeness

In considering each and every statement P defined within a formal system, the notion of consistency can be defined as "that there is no statement P which is both true and false". Similarly, the notion of completeness can be defined as "that there is no statement P which is neither true or false".

The notion of consistency is a special case of the more general concept of symmetry. To assert that a given statement P is consistent is to assert that the content, the truth or falsity of P, does not change with changes of context, the method by which P is considered (a method of derivation using other statements of the same formal system). To assert that a formal statement is 'internally consistent' is to assert that, for all statements within that formal system, that there is no statement which is true when derived by one method and then false when derived using another method. When considering the truth and falsity of a statement as its content, and the method of derivation as context, the notion of consistency is strictly equivalent to the concept of symmetry.

The notion of completeness is a special case of the more general concept of continuity. To assert that a given formal system is complete is to assert that there are no discontinuities of content. An inherent discontinuity is an implied (sharp) boundary between statements which have content, a truth or falsity value, and those which do not have a truth or falsity value because they cannot be proven by any method (using any sequence of other statements in the formal system). To assert completeness of a formal system is to assert that there is a continuity of content; that all expressions within the language of the formal system (the context) have a truth or falsity value (the content).

Insofar as the incommensuration theorem asserts that the concepts of symmetry and continuity cannot be simultaneously applied when making comparisons of statements within the language of a formal system, the concepts of absolute consistency and absolute completeness are also mutually incompatible.

Causality and Determinism

1.62-2 The concepts of causality and determinism are distinct.
They cannot be used interchangeably.

1.62-1 The notion of determinism (as per the inherent nature of the concept) is to assert that there can be a complete and absolute specification of all values of the three content metrics (pattern/mass, force, and probability) for all values of the three context metrics (time, space, and possibility).

2M33 The notion of Causality refers to persistent perceived relation of association between three events:
the observing self, the observed cause, and the observation of the effect.

1.61-10 The concept of objective causality applies to the degree that both the antecedent and the consequent are themselves objective. The concept of subjective causality applies to the degree that either the antecedent and/or the consequent is subjective.

To assert determinism is to assert that there is a fixed, exact, and well-defined microscopic structure/pattern to all eventities in a domain. Unlike the concept of causality, the concept of determinism does not require any reference to basis concepts of observability, observer, or temporality.

To assert causality is to assert a consistent pattern of observations of relations between mesoscopic eventities using the basis concepts of observer and temporality. Unlike determinism, the assertion of causality is to make no claim as to the nature, being, pattern, or degree of structure/form of the microscopic aspects of eventities in a domain.

2M34 The concept of causality requires the aspect of an observer self to be present, whereas the concept of determinism does not.

The concept of causality is time asymmetric (with respect to the observer self), whereas determinism requires no unique (non-spatial) concept of time,

The concept of causality does not require a detailed microscopic specification of all events (including that of the observer self), whereas the concept of determinism does require such specification (to be possible, at least in principle).

1.62-4 A/any/the/all worlds are to some non-zero and positive degree causal. No world/domain is purely and completely deterministic and/or logical (1).

1.62-5 Neither experience nor creativity is determined (they are not deterministic).

The (form of the) content of experience is at once proscribed by causality and described by choice.
The (form of the) content of expression (creativity) is at once proscribed by choice and described by causality.



Notes:
[1] Science is based upon observation. As such, it can only establish assertions of causality. It cannot establish or validate any claim about the determinism of any eventity. Determinism cannot be perceived or observed, and as such, any claims about absolute possibility are ultimately unscientific. Despite popular misconceptions, the natural law of science is always causal law, not ever deterministic law.


Science and Mathematics

2M35 Mathematics is organized thinking about the interactions and interdependencies of pure relations.
Mathematics is an inquiry and description of the nature of pure abstract relationship.

2M36 In this way mathematics is distinct from physics, which considers interactions (not relations) within a specific domain. Mathematics is considered in a manner which is purely omniscient modal, whereas physics is considered in a manner which also has elements of both the immanent and the transcendent modality. Science is immanent when involved in experiment (in the scientific method) and transcendent when asserting a relationship between theory and reality (a belief that a mathematical model corresponds to natural law).

2M37 Scientific knowledge is not mathematical knowledge. Scientific knowledge is based on the scientific method. The scientific method is based upon experiment (expression and perception), and measurement, whereas mathematics requires no experiment (is deterministic).

2M38 All scientific knowledge is of the causal type. There is, and there can be, no scientific knowledge of the deterministic type (due to the very nature of the scientific method itself). In that experiments are always finite (bandwidth limited), the infinite specification necessary to establish determinism can never be obtained. Insofar as experimentation involves a scientist (observer), science can only establish causal relations.


The Physical and the Non-Physical

2M27 The concept of 'the physical' refers to an existence which is both actual and deterministic. A deterministic existence is a concept that an actuality has a definite and specific form (objective content) pattern in all levels of detail (in scale) for all times and places, regardless and independent of the subjective context in which that pattern is observed.

2M28 The concept of the physical is an assertion that the total space-time structural pattern of a world is completely defined in all aspects (regardless of the frame of reference of one's perspective on it), and that this world is an actuality (for it can be perceived). In other words, the structure of the physical is completely defined at all scales and for all observers, regardless of their position in time or space or of motion through time and space; and that furthermore, this structure is objectively perceivable (i.e. is an actual observable).

2M29 The concept of 'the non-physical' refers to a creation which is both potential and non-deterministic. A non-deterministic creation is a concept of a potentiality that has a definite and specific subjective context in all levels of abstraction for all times and possibilities, but which has absolutely no specific or defined form (objective content).

2M30 The concept of the non-physical is an assertion that the total possible-time semantic meaning of a world is completely defined in all environments, regardless of the structure of one's expectations of it; and that this world is a potentiality (it cannot be perceived). The semantics of the non-physical is completely defined at all abstractions and for all expectations, regardless of the position or motion in and through time and/or possibility. The semantic of the non-physical is not objectively perceivable (i.e. it is a potentiality).


Cartesian Dualism

2M39 In distinguishing causality from determinism, there is established a strict difference between the nature of causality and the nature of the physical. Determinism and physicality are consistent with one another in that both (implicitly) involve infinite detail of (microscopic) specification. As such, the consideration of science (finite) can extend only into the nature of reality and causality. It cannot consider the nature of the physical (infinite) in any direct sense (nor can science consider creation (also infinite), nor even choice (non-visible, non-repeatable)).

2M40 In that there is a distinction between causality and the physical, there is also a distinction between choice and the non-physical. Choice is to some extent personal, and though it involves the random and dynamic, it is not absolutely chaotic. Choice is personal, whereas the pure non-physical (creation) is ultimately impersonal. In comparisons within any given domain, the energy (potentiality) involved in a choice is always finite, whereas the energy (potentiality) involved in creation is always infinite (the ratio of any number to zero (the "empty domain prior to creation") is always infinite).

2M41 To the degree that there is an assumption of interaction (finite), 1) reality is different from physicality and 2) self is different from pure non physicality/dynamicism, creation. In considering interaction, there is the idea that the self (soul), which to some extent resembles the non-physical and non-deterministic, can interact with a reality, which to some extent resembles a physical and deterministic system. The resemblance between the self (soul) and the non-physical, and the resemblance between reality and the physical, are inversely proportional to the degree of interaction assumed between self and reality.

2M42 The degree of interaction between self and reality, however, can never be zero or infinite. Therefore, interaction and consciousness are neither purely physical nor non-physical; neither deterministic nor non-deterministic, and are (to some extent) both causal and non-causal (i.e., a composition of choice).

2M43 Consciousness, in itself, belongs neither to reality nor to the self exclusively, but is shared between them in the form of interaction. Consciousness is not an illusion, nor is it in conflict with conventional science. It is both the absolute deterministic physical model and the absolute indeterministic non-physical models that are illusions.

2M44 As such, neither the realistic nor the idealistic arguments can ever be resolved by (any) logic, perception, experience, or experiment. This is due to the fact that no sensory interaction, fact of experience or objective measurement could ever encompass the infinitely small or the infinitely large equal to the presumed absolute nature of the reality itself.



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